• 소음진동
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• 제8권3호
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• pp.524-532
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• 1998

The purpose of this paper is to investigate the free vibrations of horizontally curved beams resting on Winkler-type foundations. Based on the classical Bernoulli-Euler beam theory, the governing differential equations for circular curved beams are derived and solved numerically. Hinged-hinged, hinged-clamped and clamped-clamped end constraints are considered in numerical examples. The free vibration frequencies calculated using the present analysis have been compared with the finite element's results computed by the software ADINA. Numerical results are presented to show the effects on the natural frequencies of curved beams of the horizontal rise to span length ratio, the foundation parameter, and the width ratio of contact area between the beam and foundation.

• Structural Engineering and Mechanics
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• 제15권6호
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• pp.705-716
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• 2003

Gradient elastic flexural beams are dynamically analysed by analytic means. The governing equation of flexural beam motion is obtained by combining the Bernoulli-Euler beam theory and the simple gradient elasticity theory due to Aifantis. All possible boundary conditions (classical and non-classical or gradient type) are obtained with the aid of a variational statement. A wave propagation analysis reveals the existence of wave dispersion in gradient elastic beams. Free vibrations of gradient elastic beams are analysed and natural frequencies and modal shapes are obtained. Forced vibrations of these beams are also analysed with the aid of the Laplace transform with respect to time and their response to loads with any time variation is obtained. Numerical examples are presented for both free and forced vibrations of a simply supported and a cantilever beam, respectively, in order to assess the gradient effect on the natural frequencies, modal shapes and beam response.

• Steel and Composite Structures
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• 제19권5호
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• pp.1259-1277
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• 2015

In this work, a trigonometric refined beam theory for the bending, buckling and free vibration analysis of carbon nanotube-reinforced composite (CNTRC) beams resting on elastic foundation is developed. The significant feature of this model is that, in addition to including the shear deformation effect, it deals with only 3 unknowns as the Timoshenko beam (TBM) without including a shear correction factor. The single-walled carbon nanotubes (SWCNTs) are aligned and distributed in polymeric matrix with different patterns of reinforcement. The material properties of the CNTRC beams are assessed by employing the rule of mixture. To examine accuracy of the present theory, several comparison studies are investigated. Furthermore, the effects of different parameters of the beam on the bending, buckling and free vibration responses of CNTRC beam are discussed.

• Coupled systems mechanics
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• 제6권4호
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• pp.465-485
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• 2017

In this paper, free vibration and static response of magneto-electro-elastic (MEE) beams has been investigated. To this end, a 3D finite element formulation has been derived by minimization the total potential energy and linear constitutive equation. The coupling between elastic, electric and magnetic fields can have a significant influence on the stiffness and in turn on the static behaviour of MEE beam. Further, different Barium Titanate ($BaTiO_3$) and Cobalt Ferric oxide ($CoFe_2O_4$) volume fractions results in indifferent coupled response. Therefore, through the numerical examples the influence of volume fractions and boundary conditions on the natural frequencies of MEE beam is illustrated. The study is extended to evaluate the static response of MEE beam under various forms of mechanical loading. It is seen from the numerical evaluation that the volume fractions, loading and boundary conditions have a significant effect on the structural behaviour of MEE structures. The observations made here may serve as benchmark solutions in the optimum design of MEE structures.

• 한국소음진동공학회:학술대회논문집
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• 한국소음진동공학회 2001년도 추계학술대회논문집 II
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• pp.1005-1009
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• 2001

This paper deals with the free vibration analysis of beam-columns on elastic foundation using Differential Quadrature Method. Based on the dynamic equilibrium equation of a beam element acting the stress resultants and the inertia force, the governing differential equation is derived for the in-plane free vibration of such beam-columns. For calculating the natural frequencies, this equation is solved by the Differential Quadrature Method. It is expected that the results obtained herein can be used in application of Differential Quadrature Method to the field of civil engineering and practically in the structural engineering, the foundation engineering and the vibration control fields.

• Steel and Composite Structures
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• 제46권5호
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• pp.637-647
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• 2023

Nonlinear free vibration and stability responses of a carbon nanotube reinforced composite beam under temperature rising are investigated in this paper. The material of the beam is considered as a polymeric matrix by reinforced the single-walled carbon nanotubes according to different distributions with temperature-dependent physical properties. With using the Hamilton's principle, the governing nonlinear partial differential equation is derived based on the Euler-Bernoulli beam theory. In the nonlinear kinematic assumption, the Von Kármán nonlinearity is used. The Galerkin's decomposition technique is utilized to discretize the governing nonlinear partial differential equation to nonlinear ordinary differential equation and then is solved by using of multiple time scale method. The critical buckling temperatures, the nonlinear natural frequencies and the nonlinear free response of the system is obtained. The effect of different patterns of reinforcement on the critical buckling temperature, nonlinear natural frequency, nonlinear free response and phase plane trajectory of the carbon nanotube reinforced composite beam investigated with temperature-dependent physical property.

• 대한기계학회논문집A
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• 제30권1호
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• pp.18-25
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• 2006

The purpose of this research work is to demonstrate a successful application of hybrid-mixed formulation and nodeless degrees of freedom in developing a very accurate in-plane curved beam element for free vibration analysis. To resolve the numerical difficulties due to the spurious constraints, the present element, based on the Hellinger-Reissner variational principle and considering the effect of shear deformation, employed consistent stress parameters corresponding to cubic displacement polynomials with additional nodeless degrees. The stress parameters were eliminated by the stationary condition, and the nodeless degrees were condensed by Guyan Reduction. Several numerical examples indicated that the property of the mass matrix as well as that of the stiffness matrix have a great effect on the numerical performance. The element with consistent mass matrix produced best results on convergence and accuracy in the numerical analysis of Eigenvalue problems. Also, the higher-order mixed curved beam element showed a superior numerical behavior for the free vibration analyses.

• Structural Engineering and Mechanics
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• 제58권5호
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• pp.847-868
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• 2016

In this study, the free vibration analysis of axially moving beams is investigated according to Reddy-Bickford beam theory (RBT) by using dynamic stiffness method (DSM) and differential transform method (DTM). First of all, the governing differential equations of motion in free vibration are derived by using Hamilton's principle. The nondimensionalised multiplication factors for axial speed and axial tensile force are used to investigate their effects on natural frequencies. The natural frequencies are calculated by solving differential equations using analytical method (ANM). After the ANM solution, the governing equations of motion of axially moving Reddy-Bickford beams are solved by using DTM which is based on Finite Taylor Series. Besides DTM, DSM is used to obtain natural frequencies of moving Reddy-Bickford beams. DSM solution is performed via Wittrick-Williams algorithm. For different boundary conditions, the first three natural frequencies that calculated by using DTM and DSM are tabulated in tables and are compared with the results of ANM where a very good proximity is observed. The first three mode shapes and normalised bending moment diagrams are presented in figures.

• Advances in nano research
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• 제16권3호
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• pp.251-263
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• 2024

This article investigates the free vibration behavior of carbon nanotube reinforced composite (CNTRC) beams embedded using variational analytical methods and artificial neural networks (ANN). The material properties of layered functionally graded CNTRC (FG-CNTRC) beams are estimated using nonlocal parameters modified power-law with different types of CNT distributions through the thickness direction of the beam. Adopting Eringen's nonlocal elasticity theory to capture the small size effects, the nonlocal governing equations are derived and solved using the analytical method. And also, the problem was analyzed using the ANN method. The architecture of the proposed ANN model is 3-9-1. In the experiments, we used 112 different data to predict the natural frequency using ANN. Based on the nonlocal differential constitutive relations of Eringen, the equations of motion as well as the boundary conditions of the beam are derived using Hamilton's principle. The classical beam theory is used to formulate a governing equation for predicting the free vibration of laminated CNTRC beams. According to the experimental results, the prediction ability of the ANN model is very good and the natural frequency can be predicted in ANN without attempting any experiments.

• Earthquakes and Structures
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• 제15권4호
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• pp.369-382
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• 2018

A free vibration analysis and wave propagation of functionally graded porous beams has been presented in this work using a high order hyperbolic shear deformation theory. Unlike other conventional shear deformation theories, a new displacement field that introduces indeterminate integral variables has been used to minimize the number of unknowns. The constituent materials of the beam are assumed gradually variable along the direction of height according to a simple power law distribution in terms of the volume fractions of the constituents. The variation of the pores in the direction of the thickness influences the mechanical properties. It is therefore necessary to predict the effect of porosity on vibratory behavior and wave velocity of FG beams in this study. A new function of the porosity factor has been developed. Hamilton's principle is used for the development of wave propagation equations in the functionally graded beam. The analytical dispersion relationship of the FG beam is obtained by solving an eigenvalue problem. Illustrative numerical examples are given to show the effects of volume fraction distributions, beam height, wave number, and porosity on free vibration and wave propagation in a functionally graded beam.